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Theorem supisoti 6482
Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
Hypotheses
Ref Expression
supiso.1  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
supiso.2  |-  ( ph  ->  C  C_  A )
supisoex.3  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
supisoti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
Assertion
Ref Expression
supisoti  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Distinct variable groups:    v, u, x, y, z, A    u, C, v, x, y, z    ph, u    u, F, v, x, y, z    u, R, x, y, z    u, S, v, x, y, z   
u, B, v, x, y, z    v, R    ph, v, x
Allowed substitution hints:    ph( y, z)

Proof of Theorem supisoti
Dummy variables  w  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supisoti.ti . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
21ralrimivva 2444 . . . . . 6  |-  ( ph  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )
3 supiso.1 . . . . . . 7  |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )
4 isoti 6479 . . . . . . 7  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
62, 5mpbid 145 . . . . 5  |-  ( ph  ->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
76r19.21bi 2450 . . . 4  |-  ( (
ph  /\  u  e.  B )  ->  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
87r19.21bi 2450 . . 3  |-  ( ( ( ph  /\  u  e.  B )  /\  v  e.  B )  ->  (
u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) )
98anasss 391 . 2  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  =  v  <-> 
( -.  u S v  /\  -.  v S u ) ) )
10 isof1o 5478 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  F : A -1-1-onto-> B
)
11 f1of 5157 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
123, 10, 113syl 17 . . 3  |-  ( ph  ->  F : A --> B )
13 supisoex.3 . . . 4  |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )
141, 13supclti 6470 . . 3  |-  ( ph  ->  sup ( C ,  A ,  R )  e.  A )
1512, 14ffvelrnd 5335 . 2  |-  ( ph  ->  ( F `  sup ( C ,  A ,  R ) )  e.  B )
161, 13supubti 6471 . . . . . 6  |-  ( ph  ->  ( j  e.  C  ->  -.  sup ( C ,  A ,  R
) R j ) )
1716ralrimiv 2434 . . . . 5  |-  ( ph  ->  A. j  e.  C  -.  sup ( C ,  A ,  R ) R j )
181, 13suplubti 6472 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  A  /\  j R sup ( C ,  A ,  R )
)  ->  E. z  e.  C  j R
z ) )
1918expd 254 . . . . . 6  |-  ( ph  ->  ( j  e.  A  ->  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) ) )
2019ralrimiv 2434 . . . . 5  |-  ( ph  ->  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) )
21 supiso.2 . . . . . . 7  |-  ( ph  ->  C  C_  A )
223, 21supisolem 6480 . . . . . 6  |-  ( (
ph  /\  sup ( C ,  A ,  R )  e.  A
)  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  (
j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R
z ) )  <->  ( A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) ) )
2314, 22mpdan 412 . . . . 5  |-  ( ph  ->  ( ( A. j  e.  C  -.  sup ( C ,  A ,  R ) R j  /\  A. j  e.  A  ( j R sup ( C ,  A ,  R )  ->  E. z  e.  C  j R z ) )  <-> 
( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) ) )
2417, 20, 23mpbi2and 885 . . . 4  |-  ( ph  ->  ( A. w  e.  ( F " C
)  -.  ( F `
 sup ( C ,  A ,  R
) ) S w  /\  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) ) )
2524simpld 110 . . 3  |-  ( ph  ->  A. w  e.  ( F " C )  -.  ( F `  sup ( C ,  A ,  R ) ) S w )
2625r19.21bi 2450 . 2  |-  ( (
ph  /\  w  e.  ( F " C ) )  ->  -.  ( F `  sup ( C ,  A ,  R
) ) S w )
2724simprd 112 . . . 4  |-  ( ph  ->  A. w  e.  B  ( w S ( F `  sup ( C ,  A ,  R ) )  ->  E. k  e.  ( F " C ) w S k ) )
2827r19.21bi 2450 . . 3  |-  ( (
ph  /\  w  e.  B )  ->  (
w S ( F `
 sup ( C ,  A ,  R
) )  ->  E. k  e.  ( F " C
) w S k ) )
2928impr 371 . 2  |-  ( (
ph  /\  ( w  e.  B  /\  w S ( F `  sup ( C ,  A ,  R ) ) ) )  ->  E. k  e.  ( F " C
) w S k )
309, 15, 26, 29eqsuptid 6469 1  |-  ( ph  ->  sup ( ( F
" C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350    C_ wss 2974   class class class wbr 3793   "cima 4374   -->wf 4928   -1-1-onto->wf1o 4931   ` cfv 4932    Isom wiso 4933   supcsup 6454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-isom 4941  df-riota 5499  df-sup 6456
This theorem is referenced by:  infisoti  6504  infrenegsupex  8763
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